I feel the question raised by the OP and the answers being limited to either yes or no is somehow playful. On the one hand, the question is presented as innocently objective (Objectively, is there a crisis on the US southern border? hence warranting yes or no answers) but on the other hand, there is a link to a video showing how Dems and the media changing their rhetoric on describing the situation at US southern border, in which the truth must be investigated and cannot be boiled down to simple yes or no answers.

Objectively, the mere fact that Dems and the media changed their narratives says nothing about the situation on the southern border. Therefore, an objective - non biased investigation would be required to evaluate how warranted the change in rhetoric is. Is it only to serve narrow interests? or is it due to meaningful change in relation to the conditions at the southern border that warrants describing the new situation as a "crisis".

Also the term crisis might have been used to convey two different meanings:

1- Trump describing the situation at the southern border as "crisis" refers to the number of illegal immigrants trying to enter the US, and Dems disagreeing with that by saying there is no crisis

2- The situation at the detention centers and the lack of hygiene is described as crisis by the Dems and media, and probably denied by Trump supporters.

Which meaning are you referring to in your question? not making the distinction can be a form of equivocation.

As Bundokji noted, both major parties see it as a crisis, but from a different framework.

In my opinion:

A wall or fence or barrier is not such a big deal some make it out to be. Is the U.S. a nation or is it not? Nations have borders and checkpoints when crossing from one nation to another. This is (or was) the case in Europe (before EU when I lived there) and virtually every other place. Perhaps there should be one global world government as some have advocated for in the past; but we are not there yet. There are about 200 nations on earth and nations have borders. In a recent government shut down, the dispute was over $5 billion funding Trump wanted to start the wall. The Dems refused and government was shut down for a long time. During the Bush years, the government had no problem funding that much or more per day for the Iraq War.

And then in defense of the other side; immigration from the South is not all that bad. The vast majority are coming over to seek a better life, find employment. Only a few rare cases come over with intent to do crime, receive welfare benefits. And virtually all of the jobs they seek are jobs most American (born) residents won't do; i.e., pick fruit, vegetables, landscaping, cleaning, maids, etc. The U.S. regularly receives immigrants from around the world (not just Mexico) to fill employment vacancies where there are shortages of finding qualified, educated, and trained people (for example, previous campaigns to fill nurse shortages, physicians, etc).